naginterfaces.library.stat.prob_​chisq

naginterfaces.library.stat.prob_chisq(x, df, tail='L')

prob_chisq returns the lower or upper tail probability for the ${\chi }^2$-distribution with real degrees of freedom.

For full information please refer to the NAG Library document for g01ec

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01ecf.html

Parameters

xfloat

$x$, the value of the ${\chi }^2$ variate with $\nu$ degrees of freedom.

dffloat

$\nu$, the degrees of freedom of the ${\chi }^2$-distribution.

tailstr, length 1, optional

Indicates whether the upper or lower tail probability is required.

$tail=\text{'L'}$

The lower tail probability is returned, i.e., $P(X\le x:\nu )$.

$tail=\text{'U'}$

The upper tail probability is returned, i.e., $P(X\ge x:\nu )$.

Returns

pfloat

The lower or upper tail probability for the ${\chi }^2$-distribution, depending on the value of $tail$ supplied.

Raises

NagValueError

(errno $1$)

On entry, $tail=⟨value⟩$.

Constraint: $tail=\text{'L'}$ or $\text{'U'}$.

(errno $2$)

On entry, $x=⟨value⟩$.

Constraint: $x\ge 0.0$.

(errno $3$)

On entry, $df=⟨value⟩$.

Constraint: $df>0.0$.

(errno $4$)

The series used to calculate the gamma probabilities has failed to converge. The result returned should represent an approximation to the solution.

Notes

The lower tail probability for the ${\chi }^2$-distribution with $\nu$ degrees of freedom, $P(X\le x:\nu )$ is defined by:

$$P(X\le x:\nu )=\frac{1}{2^{\nu /2}\Gamma \left(\nu /2\right)}{\int }_{0.0}^x{X}^{\nu /2-1}{e}^{-X/2}dX\text{,}x\ge 0,\nu >0\text{.}$$

To calculate $P(X\le x:\nu )$ a transformation of a gamma distribution is employed, i.e., a ${\chi }^2$-distribution with $\nu$ degrees of freedom is equal to a gamma distribution with scale parameter $2$ and shape parameter $\nu /2$.

References

NIST Digital Library of Mathematical Functions

Hastings, N A J and Peacock, J B, 1975, Statistical Distributions, Butterworth